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exsheet1 - Mathematical Methods I Natural Sciences Tripos...

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Unformatted text preview: Mathematical Methods I Natural Sciences Tripos, Part IB Dr Gordon Ogilvie Michaelmas Term 2008 Example Sheet 1 1. (a) To what quantities do the following expressions in suffix notation (using the summation convention) correspond? Simplify where appropriate. δ ii , δ ij a j , δ ij a i a j , δ ij δ ij , ǫ iji , ǫ ijk δ jk , b i ǫ ijk a k c j , ǫ ijk A 3 i A 1 k A 2 j . (b) For each of the following equations, either give the equivalent in vector or matrix notation, or explain why the equation is invalid. x i = a i b k c k + d i , x i = a j b i + c k d i e k f j , u = ( ǫ jkl v k w l ) x j , ǫ ijk x j y k ǫ ilm x l y m = 1 , A ik B kl = T ik δ kl , x α = A αi B βi y β . (c) Write the following equations in suffix notation using the summation conven- tion. ( x + μ y ) · ( x − μ y ) = 0 , x = | a | 2 b −| b | 2 a , (2 x × y ) · ( a + b ) = λ . 2. (a) Show that a · ( b × c ) = b · ( c × a ) = c · ( a × b ) . (b) Explain why ǫ ijk ǫ klm = δ il δ jm − δ im δ jl . Hence find ǫ ijk ǫ ijk . 3. A fluid flow has the constant velocity vector (in Cartesian coordinates) v ( r ) = (0 , , W ) . Explicitly calculate the volume flux of fluid, Q = integraldisplay v · d S , flowing across (a) the open hemispherical surface r = a , z greaterorequalslant 0, and (b) the disc r lessorequalslant a , z = 0. Verify that the divergence theorem holds. 4. For a surface S enclosing a volume V , apply the divergence theorem to a vector field F = a p , where a is an arbitrary constant vector and p ( r ) is a scalar field. Deduce that integraldisplay V ( ∇ p ) d V = integraldisplay S p d S ....
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